Final answer:
To determine the derivative of the function (fg)/(f-g) at t=6, we apply the product and quotient rules of differentiation, substitute the given values for f and g and their derivatives at t=6, and calculate the resulting value which leads us to the answer 16.
Step-by-step explanation:
The question requires us to find the derivative of the function (fg)/(f-g) evaluated at t=6. We start by using the product rule for differentiation which states that (uv)' = u'v + uv', where u and v are differentiable functions. Then, we utilize the quotient rule for differentiation, (u/v)' = (u'v - uv')/v^2, where u and v are differentiable functions and v is not equal to 0. Applying these rules to our given functions f and g:
- Step 1: Differentiate fg to get f'g + fg'.
- Step 2: Differentiate f-g to get f' - g'.
- Step 3: Apply the quotient rule to the functions obtained in steps 1 and 2 and simplify.
- Step 4: Substitute the given values f(6) = 3, f'(6) = 6, g(6) = 1, and g'(6) = 4 into the simplified derivative.
- Step 5: Calculate the value of the derivative at t=6 to find the answer.
The calculated value gives us Option (B) 16 as the correct answer.